PAT 2021
1. What is the next number in the sequence $1,32,243,1024,3125$? This is obviously not an arithmetic sequence. Next choice is a geometric sequence that does not work. However we notice that $1=1^5$, $32=2^5$, $243=3^5$, $1024=4^5$ and $3125=5^5$ so the sequence is $a_n=n^5$ with the next element $6^5=7776$. 2. What is the effective spring constant of the combination of springs shown in the diagram, if each spring has spring constant $k$? 3. Evaluate $\sum_{n=1}^{10} \qty(2-\frac{n}{2}+2^n)$. Sum of arithmetic series: $\sum_{n=1}^N n=\frac{1}{2}N(N+1)$; sum of geometric series $\sum_{n=1}^N r^n=\frac{r\qty(1-r^n)}{1-r}$. $$ \begin{align*} \sum_{n=1}^{10} 2 & = 20 \\ \sum_{n=1}^{10} \frac{n}{2} & = \frac{1}{4} 10 \times 11 = \frac{55}{2} \\ \sum_{n=1}^{10} 2^n & = 2\qty(2^{10}-1) \end{align*} $$ Therefore $$ \sum_{n=1}^{10} \qty(2-\frac{n}{2}+2^n)=2^...